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4. Using a truth table, determine whether the following is true: A A ¬A - B. Then briefly explain what it is about this truth table that either shows that it is true, or shows that it is false.

4. Using a truth table, determine whether the following is true: A A ¬A - B. Then briefly explain what it is about this truth table that either shows that it is true, or shows that it is false.

Database System Concepts
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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### Question 4: Using a truth table, determine whether the following is true: \( A \land \neg A \models B \). Then briefly explain what it is about this truth table that either shows that it is true, or shows that it is false. --- **Explanation:** To solve this problem, you need to create a truth table to evaluate the expression \( A \land \neg A \models B \). 1. **Truth Table Construction:** - Create columns for each part: \( A \), \( \neg A \), \( A \land \neg A \), and \( B \). - Evaluate \( A \land \neg A \) for all possible truth values of \( A \). - Analyze whether the implication holds for each combination of truth values for \( A \) and \( B \). 2. **Analysis:** - The expression \( A \land \neg A \) always evaluates to false because one cannot be true while the other is also true simultaneously (they are logical contradictions). - Due to this contradiction, \( A \land \neg A \) is always false, making \( A \land \neg A \models B \) vacuously true for any \( B \). The key point of the truth table is that since \( A \land \neg A \) is never true, the outcome \( B \) does not affect the validity of the implication.
Transcribed Image Text:### Question 4: Using a truth table, determine whether the following is true: \( A \land \neg A \models B \). Then briefly explain what it is about this truth table that either shows that it is true, or shows that it is false. --- **Explanation:** To solve this problem, you need to create a truth table to evaluate the expression \( A \land \neg A \models B \). 1. **Truth Table Construction:** - Create columns for each part: \( A \), \( \neg A \), \( A \land \neg A \), and \( B \). - Evaluate \( A \land \neg A \) for all possible truth values of \( A \). - Analyze whether the implication holds for each combination of truth values for \( A \) and \( B \). 2. **Analysis:** - The expression \( A \land \neg A \) always evaluates to false because one cannot be true while the other is also true simultaneously (they are logical contradictions). - Due to this contradiction, \( A \land \neg A \) is always false, making \( A \land \neg A \models B \) vacuously true for any \( B \). The key point of the truth table is that since \( A \land \neg A \) is never true, the outcome \( B \) does not affect the validity of the implication.
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Introduction :

Given Expression :

  • A ^ ~A |= B
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